How to characterize flasque sheaves in more functorial way?

Question

The motivation to ask this question is some proposition of flasque sheaves. Let's recall the definition of flasque sheaf:A sheaf $F$ on a topological space $X$ is flasque if for every inclusion $V\rightarrow U$ of open sets,the restriction map $F(U)\rightarrow F(V)$ is surjective.

It is known that the proposition of flasque sheaves is very similar to injective sheaves.For example,higher cohomology of flasque sheaves vanishes.On the noetherian topological space,direct limit of flasque sheaves is flasque.The category of quasi coherent sheaves on noetherian schemes has enough flasques.And we have localization of flasque sheaves is flasque.Notice that since any injective $O_X$-module is flasque,then injective sheaves have all the propositions BUT the propositions for injectives can be proved independently without knowing they are flasque.

Notice that the injective sheaf is injective object of category of sheaves of $O_X$-modules.So it's defined purely in terms of objects and morphism in this category without mentioning the section of this sheaf.But the definition of flasque sheaf is defined in the terms of certain properties of sections of this sheaf.So My question is

1. Is there any definition of flasque sheaf in the category $O_X-mod$ only in terms of objects and morphisms(section FREE)?

2. Notice that the definition of injective object in certain category actually depends on the class of morphisms we choose.For example,in the abelian category,the standard definition of injective objects is a special case of $E$-injectives where $E$ is class of all the monomorphisms in this category.We can of course formulate some other "non-standard" injectives by choosing other class of morphisms. For example,in the category of commutative affine schemes,we can choose $E$ to be the class of proper monomorphisms of affine schemes such that the correspondence surjective map between two commutative rings having square zero kernel.Then the $E$-injective objects in category of commutative affine schemes is precisely the smooth affine schemes in this category.Therefore,Is it possible that the flasque sheaves are certain "non standard" $E$-injectives in category $O_X-mod$ with $E$ to be some class of morphisms in this category(We also notice that the class of morphisms E can be viewed as Grothendieck pretopology in this category,but maybe not finest).

If it is the case,then all the proposition of flasque sheaves can be proved purely general abstract just like injective sheaves.

I am not sure whether it is a stupid question,maybe it is known but I just did not get the right place to look at.Thanks

Answer 1

Yes — by the Yoneda lemma, flasque sheaves can indeed be seen as $E$-injectives, where $E$ consists of the inclusion maps $\newcommand{\O}{\mathcal{O}}\newcommand{\restr}{\mathord{\upharpoonright}}\O_X\restr_U \to \O_X\restr_V$, for all pairs of opens $U \subset V$.

Here $\O_X \restr_U$ is the sheaf of modules with $\O_X\restr_U(U')$ taken to be $\O_X(U')$ if $U' \subseteq U$, and to be $0$ otherwise. This is the free $\O_X$-module on the Yoneda sheaf of sets $\newcommand{\y}{\mathbf{y}}\y(U)$, defined by $\y(U)(U') = 1$ if $U' \subseteq U$ and $\emptyset$ otherwise. By the Yoneda lemma, for any sheaf of sets $F$, $F(U)$ is isomorphic to the set of sheaf maps $\y(U) \to F$. For $F$ a sheaf of modules, maps $\y(U) \to |F|$ (where $|F|$ is the underlying sheaf of sets of $F$) correspond to module maps $\O_X\restr_U \to F$; so elements of $F(U)$ correspond to such maps, and restriction $F(V) \to F(U)$ corresponds to composition with the inclusion map $\O_X\restr_U \to \O_X \restr_V$. So this restriction map is surjective exactly if $F$ is injective w.r.t. that inclusion map; and $F$ is flasque exactly if it’s injective w.r.t. the set of all such inclusions.

Answer 2

Let $\mathcal O$ be a sheaf of rings over $X$ and let $\mathcal O$-$\mathop{mod}$ be the category of $\mathcal O$-modules.

Because one has the equivalence $$\Gamma(U;\_)=\mathop{\rm Hom}\nolimits_{\mathcal O}(\mathcal O_U,\_)$$ for all open $U\subset X$, an $\mathcal O$-module $\mathcal M$ is a flabby sheaf, if and only if, the natural homomorphism $$\mathop{\rm Hom}\nolimits_{\mathcal O}(\mathcal O,\mathcal M)\to\mathop{\rm Hom}\nolimits_{\mathcal O}(\mathcal O_U,\mathcal M)\qquad (\ddagger)$$ is surjective. Question 1 should then be rephrased as whether it is possible to characterize the class of sheaves $\{\mathcal O_U\}$ in the category $\mathcal O$-$\mathop{mod}$ only by categorical means (i.e. without explicitly referring to the topology of $X$).

Exercise II.10 (p. 133) of Kashiwara-Schapira's book "Sheaves on manifolds", says that if $\mathcal O$ is a sheaf of rings over $X$, an $\mathcal O$-module $\mathcal M$ is injective in the category $\mathcal O$-$\mathop{mod}$ if and only if, for every $\mathcal O$-submodule (ideal) $\mathcal J\subset\mathcal O$ the natural homomorphism $$\mathop{\rm Hom}\nolimits_{\mathcal O}(\mathcal O,\mathcal M)\to\mathop{\rm Hom}\nolimits_{\mathcal O}(\mathcal J,\mathcal M)\qquad (*)$$ is surjective.

These presentations $(\ddagger)$ and $(*)$, that have the advantage that they highlight the "gap" between flabbies and injectives, have the disadvantage that $(\ddagger)$ refers to the sheaves $\mathcal O_U$ which may be inexistant in the category you're interested on. For exemple, if one is interested in the category $\mathop{mod}\nolimits_{\rm q.c.}(\mathcal O)$ of quasi-coherent sheaves over affine schemes, the sheaves $\mathcal O_U$ are not objets of $\mathop{mod}\nolimits_{\rm q.c.}(\mathcal O)$.

Now, in the case of the category $\mathcal O$-$\mathop{mod}$ of $\mathcal O$-modules, if the implication injective$\Rightarrow$flabby is obvious, the converse may also be truth. For example, whenever any ideal $\mathcal J\subset \mathcal O$ is isomorphic to some $\mathcal O_U$. This is in fact the case if $\mathcal O$ is the constant sheaf $k_X$, where $k$ is a field (loc. cit.). In this case, obviously, the characterization of flabby is categorical, because they are injective, but also because the class of sheaves $\{\mathcal O_U\}$ is the class of kernels of morphisms in $\mathcal O$-$\mathop{mod}$ with source $\mathcal O$.